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Question Number 193512 by SaRahAli last updated on 15/Jun/23

Answered by Subhi last updated on 15/Jun/23

$$\int \frac{1}{\frac{1}{cos\left(x\right)}.\frac{sin\left(x\right)}{cos\left(x\right)}}\Rrightarrow \int \frac{{cos}^2\left(x\right)}{sin\left(x\right)}.dx$$$$\int \frac{1-{sin}^2\left(x\right)}{sin\left(x\right)}.dx\looparrowright \int csc\left(x\right)+\int -sin\left(x\right)$$$$\int csc\left(x\right)+cos\left(x\right)$$$$\int csc\left(x\right).\frac{csc\left(x\right)+cot\left(x\right)}{csc\left(x\right)+cot\left(x\right)}+cos\left(x\right)$$$$-\int \frac{-csc\left(x\right)cot\left(x\right)-{csc}^2\left(x\right)}{csc\left(x\right)+cot\left(x\right)}+cos\left(x\right)\Rrightarrow -ln\mid csc\left(x\right)+cot\left(x\right)\mid +cos\left(x\right)$$

Commented by MM42 last updated on 15/Jun/23

$$⊛\int \frac{dx}{sinx}=\int \frac{1+{tan}^2\frac{x}{2}}{2tan\frac{x}{2}}dx=ln\left(tan\frac{x}{2}\right)+c$$

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